COMMUTING CONTRACTIONS の SIMULTANEOUS UNITARY DILATION(Linear Operators and Inequalities)

DILATN13.TEX; 1994/01/31

COMMUTING

CONTRACTIONS

SIMULTANEOUS UNITARY

DILATION

山形大学理学部 岡安隆照 (Takateru OKAYASU)

The following matter is realy fundamental:

Sz.-Nagy’s Unitary Dilation Theorem. Let $T$ be a contraction on a

Hilbert space $\mathcal{H}$. Then, there exist an enlarged Hilbert space $\mathcal{K}\supseteq \mathcal{H}$ and a

unitary $U$, called a unitary dilation of $T$, on $\mathcal{K}$, such that

$T^{m}=PU^{m}|\mathcal{H}$ for $m=0,1,2,$ $\cdots$ ,

where $P$ is the projection on $\mathcal{K}$ onto $\mathcal{H}$

.

This yields, and, is yielded by, the so-called

von

Then,

$||p(T)|| \leq||p||=\sup_{z\in T}|p(z)|$

holds for any polynomial $p$ with complex coeflicients.

Theorem [6]. If a set of commuting contractions on a Hilbert space $\mathcal{H}$,

$T_{1},$ $T_{2},$

$\cdots,$$T_{n}$, admits a simultaneous unitary dilation, namely, there exist a

Hilbert space $\mathcal{K}\supseteq \mathcal{H}$ and commuting unitaries $U_{1},$ $U_{2)}\cdots,$ $U_{n}$ on $\mathcal{K}$, such

that

$T_{1}^{m_{1}}T_{2}^{m_{2}}\cdots T_{n}^{m_{n}}=PU_{1}^{m_{1}}U_{2}^{m_{2}}\cdots U_{n}^{m_{n}}|\mathcal{H}$

for $m_{1},$ $m_{2},$ $\cdots,$ $m_{n}=0,1,2,$ $\cdots$, where $P$is the projection on

$\mathcal{K}$ onto $\mathcal{H}$, then

$T_{1)}T_{2},$ _$\cdots,$$T_{n}$ enjoys the von Neumann ineqality, namely,

$||(p_{i}j(T_{1}, T_{2}, \cdots, T_{n}))||\leq||(p_{ij})||=\sup_{z_{1)}z_{2))}z_{n}\in T}||(p_{ij}(z_{1)}z_{2)}\cdots, z_{n}))||$

holds for any $m\cross m$ matrix $(p_{ij})$ whose entries are polynomials with complex

coeffieients; and vice versa.

On the other hand, the following theorems are known:

And\^o’s Theorem [1]. Any pair of commuting contractions on a Hilbert

space admits a simultaneous unitary dilation.

And\^o’s Theorem [2]. Any triple ofcommuting contractions on aHilbert

space, one of which duoble commutes with others, admits a simultaneous

unitary dilation.

We, aside, have examples of triples of commuting contractions which do

not admit a simultaneous unitary dilation, [4], [8] and [9]. In [6] we

gave

Theorem. Suppose each of sets of commuting contractions, $S_{1}\rangle$ _{$S_{2)}\cdots$},

$S_{m}$ and $T_{1},$$T_{2,}T_{n}$, on a Hilbert space, admits a simultaneous unitary

dilation, and every $S_{j}$ double commutes with all $T_{k}$

.

generates a nuclear $C^{*}$ algebra, then the set $S_{1},$ $S_{2},$ _$\cdots,$ $S_{m},$ $T_{1},$ $T_{2},$ _$\cdots,$ $T_{n}$

admits a simultaneous unitary dilation.

Collorary. Suppose $S$ is a $GCR$ contraction, i.e., a contraction which

generates a $GCR$ (postliminal) algebra, $T_{1},$ $T_{2},$

$\cdots,$$T_{n}$ commuting

dilation. and $S$ double commutes with all $T_{k}$. Then the set $S,$$T_{1},$ $T_{2},$ _$\cdots,$$T_{n}$

admits a simultaneous unitary dilation.

The following, furtheremore, turned out to be true [7]:

Theorem. Suppose each of sets of commuting contractions, $S_{1},$ $S_{2}$,

$\ldots,$ $S_{m}$ and $T_{1},$ $T_{2},$ $\cdots,$ $T_{n}$, on a Hilbert space, admits a simultaneous

unitary dilation, and every $S_{j}$ double commutes with all $T_{k}$. If the set

$S_{1},$ $S_{2},$

$\cdots,$ $S_{m}$ generates an injective von Neumann algebra, then the set

$S_{1},$ $S_{2},$ _$\cdots,$ $S_{m},$$T_{1},$$T_{2,}T_{n}$ admits a simultaneous unitary dilation.

Collorary. Suppose $S$ is a type I contraction, i.e., a contraction which

generates a type I von Neumann algebra, $T_{1},$$T_{2},$

$\cdots,$ $T_{n}$ commuting

contrac-tions, on a Hilbert space, the set $T_{1},$$T_{2},$

$\cdots,$ $T_{n}$ admits a simultaneous unitary

dilation and $S$ double commutes with all $T_{k}$. Then, the set $S,$$T_{1)}T_{2},$ _$\cdots,$$T_{n}$

admits a simultaneous unitary dilation.

We here will improve the theorem, by making the assumption thin as the

following

Theorem. Suppose each of sets of commuting contractions, $S_{1},$ $S_{2}$,

$\ldots,$ $S_{m}$ and $T_{1},$$T_{2},$ $\cdots,$$T_{n}$, on a Hilbert space, admits a simultaneous

uni-tary dilation, and every $S_{j}$ double commutes with all $T_{k}$

.

$S_{1},$ $S_{2},$ $\cdots$ , $S_{m},$ $T_{1},$ $T_{2},$ $\cdots$ , $T_{n}$ admits a simultaneous unitary dilation.

This is the aimed theorem of ours. A proof of this is given, on acount of

the Steinspring representation of completely positive maps, by the preceding theorem and the

Arveson Theorem [3, Theorem 1.3.1]. Let $\mathcal{H},$ $\mathcal{K}$ be Hilbert spaces,

$V$ a bounded operator from $\mathcal{H}$ into $\mathcal{K}$, and $\mathcal{B}$ a *subalgebra of $\mathcal{B}(\mathcal{K})$, the

full operator algebra, which satisfies that $[BV\mathcal{H}]=\mathcal{K}$

.

$T\in(V^{*}\mathcal{B}V)’$ there exists a unique $\tilde{T}\in \mathcal{B}’$ such that $\tilde{T}V=VT$, and the

We have as well

Collorary. Suppose each of pairs of commuting contractions, $S_{1},$ $S_{2}$, and

$T_{1},$$T_{2}$, on a Hilbert space, admits a simultaneous unitary dilation, and each

of $S_{1},$ $S_{2}$ double commutes with $T_{1}T_{2}\rangle$ Then, the set $S_{1}S_{2},$$T_{2}\rangle$$T_{1},$ admits a

simultaneous unitary dilation.

Our theorem, of course, gives a good understanding to And\^o’s “triple“

assertion; on the And\^o’s “pair“ assertion, the next matter sheds light:

Theorem [5, Theorem 6]. Let $T$ be a contraction on a Hilbert space $\mathcal{H}$,

$U$ the minimal unitary dilation of $T$

.

$\tilde{S}\in\{U\}’$ such that $S=P\tilde{S}|\mathcal{H}$ and $||\tilde{S}||=||S||$

.

References

1. T. And\^o, On a pair of commuting contractions, Acta Sci. Math.

24(1963), 88-90.

2. T.

And\^o,

Acad. Polonaise Math. 24(1976), 851-853.

3. W. B. Arveson, Subalgebras of $C^{*}$ algebras, Acta Math. 123(1969),

141-224.

4. M. J. Crabb, and A. M. Davie, von Neumann’s inequality for Hilbert

space operators, Bull. London Math. Soc. 7(1975), 49-50.

5. R. G. Douglas, P. S. Muhley, and C. Pearcy, Lifting commuting

oper-ators, Michgan Math. Journ. 15(1968), 385-395.

6. T. Okayasu, The von Neumann inequality and dilation theorems for

contractions, Operator Theory: Adv. Appl. 59(1992), 285-291.

7. 岡安隆照, The von Neumann inequality, 日本数学会秋期総合分科会)

8. I. Parrott, Unitary dilations for commuting contractions, Pacif. Jorn.

Math. 34(1970), 481-490.

9. N. Th. Varopoulos, On an inequality of von Neumann and an appli-cation of metric theory of tensor products to operator theory, Journ.